MOTION OF A PARTICLE 289
4-Vi/v^ d^uf dv dv
tnen = - = -v
dd2 dd dur '
and v^L + u' = 0
du
Separating the variable and integrating
i^/ = ^v____ Integrating again
ni-
cos"1 - = e + 8
A
or uf = A cos
Let u' = A when 0 =0, then 5 = 0. Therefore
u' = A cos 0 (3)
is a solution of equation (2). Substituting the value of uf in equation (3),
u=f- + Acos6 (4)
Ai"
and replacing u by its value
r=-----^ (5)
1 e cos 6
h2A 1
where e =-------, p = - , (6)
At A
Equation (5) is the well-known equation of a conic section. Therefore the orbit is a conic section with an eccentricity
i * h*A equal to------
M The expression for the velocity at any point" of the orbit
C!T dd may be obtained by substituting the values ofand . which